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As the variable
is replaced with in the integral

each value of in the integral
needs to be associated with the appropriate value of the field
which is
.The new field represents
a remapping of the original field .Each value in the new field with
coordinates corresponds to the value
in the field with coordinates
.
The remapping can be understood easier
if one takes the particular case *k*_{y}=0.
The change of variable in (5) becomes

the Jacobian
and the zero-offset migrated field becomes
| |
(10) |

Compare equation (10) with the Gazdag (1978) and
Stolt (1978) zero-offset migration equations:
| |
(11) |

Notice that the Stolt migration formula is comprised of an
inverse Fourier transform and a remapping (interpolation).
Equation (10) inside the integral in *k*_{h}
has the same form
as the remapping in the Stolt formula, save for a constant
coefficient . This similitude suggests the equivalent
formulation:
| |
(12) |

The validity of the new equation is verified
by replacing in equation (12) the
exponential expression
by a new variable to get back the inverse Fourier transform
of the initial expression.
The integration in *k*_{h} represents the inverse Fourier
transform at zero offset (*h*=0). However because
equation (12) actually performs summation along
hyperbolas in the spatial coordinate *h*, replacing
the integral in *k*_{h} by the inverse Fourier transform will not
return the constant-offset sections.

Applying the same technique to the case I obtain

| |
(13) |

which can be verified by substituting the exponential
expression in with a new variable defined as
| |
(14) |

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Stanford Exploration Project

11/17/1997